Even if we are interested in all categories, an alternative objective is to maximize the average precision of category pair comparisons (see Sec. 5A.13 in [1]}), which is equivalent to minimizing the sum
\( \mathbb {V}_\Sigma =\sum _i \mathbb {V}(\hat{\mu }_i)=\sum _i \frac{\sigma _i^2}{n_i}.\) 
In this case, proportional allocation achieves
\(\mathbb {V}_\Sigma ^{\scriptscriptstyle \textrm {prop}}= \frac{N}{n} \sum _i \frac{\sigma _i^2}{|C_i|}\) .
while, using Lagrange multipliers we get
\(n_i^{\scriptscriptstyle \textrm {opt}}= \frac{\sigma _i}{\sum _j \sigma _j}\cdot n \qquad \textrm { and } \qquad \mathbb {V}_\Sigma ^{\scriptscriptstyle \textrm {opt}}= \frac{\left(\sum _i \sigma _i\right)^2}{n},\) 
